Let f: R → R be a function given by f(x) = { {1-cos2x / x^2 ; x < 0}, {α, x = 0}, {β√1-cosx / x} , x > 0

Question

Let \(f: \mathrm{R} \rightarrow \mathrm{R}\) be a function given by

\(f(x)=\left\{\begin{array}{cl} \frac{1-\cos 2 x}{x^{2}} & , x<0 \\ \alpha & , x=0, \text { where } \alpha, \beta \in R. \\ \frac{\beta \sqrt{1-\cos x}}{x} & , x>0 \end{array}\right.\)

If \(f\) is continuous at \(\mathrm{x}=0\), then \(\alpha^{2}+\beta^{2}\) is equal to :

(1) 48

(2) 12

(3) 3

(4) 6

Answer 1

Correct option is (2) 12

\(f\left(0^{-}\right)=\lim \limits_{x \rightarrow 0^{-}} \frac{2 \sin ^{2} x}{x^{2}}=2=\alpha\)

\(f\left(0^{+}\right)=\lim\limits _{x \rightarrow 0^{+}} \beta \times \sqrt{2} \frac{\sin \frac{x}{2}}{2 \frac{x}{2}}=\frac{\beta}{\sqrt{2}}=2\)

\(\Rightarrow \beta=2 \sqrt{2}\)

\(\alpha^{2}+\beta^{2}=4+8=12\)